Finding tan(pi/8) | How Many Ways Are There?

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❤️ tan(pi/8)

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[justabunga]

You can use the half angle formula if you rewrite tan(π/8)=tan((1/2)(π/4)). Since there are three tangent angle formulas, all three have the correct answer. The formula for those are In order to get only one answer, we have to get rid of the ± sign. What we know is that π/8 is in the first quadrant, so we can reject the negative sign. To make things easier, I will go ahead and use the formula We can clear the complex fraction by multiplying both the top and bottom by √(2), which is √(2)-1, or -1+√(2). If you want, you can also check the other 2 equations and will lead the same answer as before.

[AbouTaim-Lille]

tg² Θ +1 = 1/cos² Θ
So U need to calculate cos π/8.
As you know cos2Θ = 2 cos² Θ -1.
Thus cos² π/8 = ½(cos π/4 +1 )
= ½ (√2+1)
So tg² π/8 = 2/(√2+1) - 1.
The rest is just calculus.

[wryanihad]

4th method

Tan(x)=sin(x)/[1+cos(x)]
Put x=π/8 then
tan(π/8)=sqr(2)-1

[kassuskassus]

Very nice ! You are the Ethan Hunt of the mathematics, sir !☺☺☺

[Chrisoikmath_]

2nd method is very smart and excellent!! You actually seldom use geometry to solve problems because you do not like it very much.
My method: In triangle ABC you can take the bisector of B (let it be BD) and use the bisector theorem: AB/AC=AD/CD to find tan(π/8). 😊

[NadiehFan]

When at 10:23 you have the system

a² − b² = 1
2ab = 1

you can use the identity (a² + b²)² = (a² − b²)² + 4a²b² to get

a² + b² = √2

and from this and a² − b² = 1 we easily find

a² = ½(√2 + 1)
b² = ½(√2 − 1)

This gives b²/a² = (√2 − 1)/(√2 + 1) = (√2 − 1)² and since tan ⅛π = b/a > 0 we immediately get tan ⅛π = √2 − 1.

Fourth method: simply use the tangent half angle formula

tan ½θ = sin θ/(1 + cos θ)

and the known values sin ¼π = cos ¼π = ½√2 to get

tan ⅛π = √2/(2 + √2) = √2·(2 − √2)/2 = (2√2 − 2)/2 = √2 − 1.

[anthonycheng]

Use tan(x/2) = sin(x/2)/cos(x/2) = = sqrt((1-cos(x))/(1+cos(x)), sqrt is positive as x is acute.

[petrileskinen]

My method of solving this might be a bit hard to explain with words but it is using the unit circle, a bit similar to your second method.
First draw an angle of pi/4 in the first quadrant, so that it ends at the point (1/sqrt(2), 1/sqrt(2)).
Next, connect that point on the circle to the point (-1, 0) remembering that the central angle is pi/4, and there for the inscribed angle at (-1, 0) becomes a half of that: pi/8 e.g. what we were looking after.
Therefore, looking at the angle at (-1, 0), the opposite side has length 1/sqrt(2), and the adjacent side has lenght 1+1/sqrt(2).
Finally we can calculate tan(pi/8) = (1/sqrt(2)) / (1+1/sqrt(2)) = ... = sqrt(2)-1 ... and this brings me to the end of my comment.

[wasimvillidad]

I often ask these two questions together in my mathematics classes - what is tan(π/8) and what is the mass of an 80 gsm A4 sheet of paper? If you know, you know.

[roberttelarket]

Not 2t since I hate tea. To coffee definitely.

[RealQuInnMallory]

{tanpi+e^xLoge^xtan8 ➖ }= { } = tanpi4^2.(tanpie^x ➖ 4/tanpi +2

[roberttelarket]

Not 2t since I hate tea. To coffee definitely.